File size: 1,575 Bytes
f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | import Mathlib
import OAI.Probability.CriticalPercolation.Exploration.ClosedPivotWeight
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
open scoped Classical
namespace CriticalPercolation.LazyDFS
open scoped Classical
open FiniteProduct SearchTree
variable {V E : Type*} [Fintype E] [Fintype V] [DecidableEq V] [DecidableEq E]
theorem search_marked_contact_bound (G : BondGraph V E) (B : Set V)
(order : List (Dart V E)) (horder : GoodDarts G order) (ho : Exhaustive G order)
(a : V) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) (w : E → Prop → ℝ)
(h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p) :
let F := fun ω : Set E => ∃ b ∈ B, (G.openGraph ω).Reachable a b
let T := search G order a B
expect w (fun ω => ∑ e, T.closedPivotWeight p F w ω e *
(1 + T.prefixScore p F w (Function.update ω e True) e)) ≤ 4 := by
let F := fun ω : Set E => ∃ b ∈ B, (G.openGraph ω).Reachable a b
let T := search G order a B
have hv : T.Valid ∅ := compile_valid G order B _ _ _
have hF : ∀ s t, s ⊆ t → F s → F t := by
rintro s t hst ⟨b,hb,hab⟩
exact ⟨b,hb,hab.mono (G.openGraph_mono hst)⟩
have hm := search_target_pivotal_moments G B order horder ho a hp w h0 h1
have hb := marked_prefix_bound T ∅ hv hp F hF w (homogeneous_normalized w p h0 h1)
(fun e b => (homogeneous_positive w hp h0 h1 e b).le)
(fun e _ => ⟨h0 e,h1 e⟩) h0 h1
exact hb.trans (by linarith [hm.1,hm.2])
end CriticalPercolation.LazyDFS
end
end
end OAI
|